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* @license Apache-2.0
*
* Copyright (c) 2025 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var abs = require( '@stdlib/math/base/special/fast/abs' );
// MAIN //
/**
* Computes the scalar multiple of the first column of `K` where `K = (H - Z1)*(H - Z2)` for a 2-by-2 or a 3-by-3 matrix `H` and where `Z1 = z1*I`, `Z2 = z2*I`, `z1 = a + bi`, `z2 = c + di`, and `I` is the identity matrix.
*
* ## Notes
*
* - It is expected that either `sr1 = sr2` and `si1 + si2 = 0` or `si1 = si2 = 0` (i.e., they represent complex conjugate values).
* - This is useful for starting double implicit shift bulges in the QR algorithm.
* - `V` should have at least `N` indexed elements.
*
* @private
* @param {PositiveInteger} N - number of row/columns in `H`
* @param {Float64Array} H - input matrix
* @param {integer} strideH1 - stride of the first dimension of `H`
* @param {integer} strideH2 - stride of the second dimension of `H`
* @param {NonNegativeInteger} offsetH - index offset for `H`
* @param {number} sr1 - real part of the first conjugate complex shift
* @param {number} si1 - imaginary part of the first conjugate complex shift
* @param {number} sr2 - real part of the second conjugate complex shift
* @param {number} si2 - imaginary part of the second conjugate complex shift
* @param {Float64Array} V - output array
* @param {integer} strideV - stride length for `V`
* @param {NonNegativeInteger} offsetV - index offset for `V`
* @returns {Float64Array} `V`
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* var H = new Float64Array( [ 1.0, 3.0, 2.0, 2.0, 4.0, 6.0, 0.0, 5.0, 7.0 ] ); // => [ [ 1.0, 3.0, 2.0 ], [ 2.0, 4.0, 6.0 ], [ 0.0, 5.0, 7.0 ] ]
* var V = new Float64Array( 3 );
*
* var out = dlaqr1( 3, H, 3, 1, 0, 1.5, 0.0, 2.5, 0.0, V, 1, 0 );
* // returns <Float64Array>[ ~1.93, ~0.57, ~2.86 ]
*/
function dlaqr1( N, H, strideH1, strideH2, offsetH, sr1, si1, sr2, si2, V, strideV, offsetV ) { // eslint-disable-line max-params, max-len
var h21s;
var h31s;
var h11;
var h12;
var h13;
var h21;
var h22;
var h23;
var h31;
var h32;
var h33;
var iv;
var s;
var i;
h11 = offsetH;
h12 = offsetH + strideH2;
h21 = offsetH + strideH1;
h22 = h21 + strideH2;
if ( N === 2 ) {
s = abs( H[ h11 ] - sr2 ) + abs( si2 ) + abs( H[ h21 ] );
if ( s === 0.0 ) {
V[ offsetV ] = 0.0;
V[ strideV + offsetV ] = 0.0;
return V;
}
h21s = H[ h21 ] / s;
V[ offsetV ] = ( h21s * H[ h12 ] ) + ( ( H[ h11 ]-sr1 ) * ( ( H[ h11 ]-sr2 ) / s ) ) - ( si1*( si2 / s ) ); // eslint-disable-line max-len
V[ offsetV + strideV ] = h21s*( H[ h11 ]+H[ h22 ]-sr1-sr2 );
return V;
}
h13 = h12 + strideH2;
h31 = h21 + strideH1;
h33 = h22 + strideH1 + strideH2;
h23 = h22 + strideH2;
h32 = h22 + strideH1;
s = abs( H[ h11 ]-sr2 ) + abs( si2 ) + abs( H[ h21 ] ) + abs( H[ h31 ] );
if ( s === 0.0 ) {
iv = offsetV;
for ( i = 0; i < 3; i++ ) {
V[ iv ] = 0.0;
iv += strideV;
}
return V;
}
h21s = H[ h21 ] / s;
h31s = H[ h31 ] / s;
iv = offsetV;
V[ iv ] = (( H[ h11 ]-sr1 )*( ( H[ h11 ]-sr2 ) / s )) - (si1*( si2 / s )) + ((H[ h12 ]*h21s) + (H[ h13 ]*h31s)); // eslint-disable-line max-len
iv += strideV;
V[ iv ] = (h21s*( H[ h11 ]+H[ h22 ]-sr1-sr2 )) + (H[ h23 ]*h31s);
iv += strideV;
V[ iv ] = (h31s*( H[ h11 ]+H[ h33 ]-sr1-sr2 )) + (h21s*H[ h32 ]);
return V;
}
// EXPORTS //
module.exports = dlaqr1;
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